Question

Reduce \(\frac{144}{216}\) to its lowest terms. A. \(\frac{18}{27}\) B. \(\frac{2}{3}\) C. \(\frac{36}{54}\) D. \(\frac{72}{108}\)

Short answer

Option B: \(\frac{2}{3}\)

Step-by-step solution

Identify the Greatest Common Divisor (GCD)

To reduce the fraction \(\frac{144}{216}\) to its lowest terms, find the Greatest Common Divisor (GCD) of 144 and 216. The GCD of 144 and 216 is 72.

Divide the Numerator and Denominator by the GCD

Divide both the numerator (144) and the denominator (216) by their GCD (72): \[ \frac{144 \div 72}{216 \div 72} = \frac{2}{3} \]

Verify the Result

The resulting fraction \(\frac{2}{3}\) is already in its lowest terms because 2 and 3 have no common divisors other than 1.

Match with Given Options

Compare the reduced fraction \(\frac{2}{3}\) with the given options: \(\frac{18}{27}\), \(\frac{2}{3}\), \(\frac{36}{54}\), \(\frac{72}{108}\). The correct option is B. \(\frac{2}{3}\).

Key concepts

finding the greatest common divisor

To reduce a fraction, the first step is to find the Greatest Common Divisor (GCD) of the numerator and the denominator. The GCD is the largest number that can divide both numbers without leaving a remainder. For example, to reduce the fraction \(\frac{144}{216}\), we need to find the GCD of 144 and 216.
Common methods for finding the GCD include listing out the factors of each number and using the Euclidean algorithm.
After finding the factors:
- Factors of 144: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
- Factors of 216: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216

The common factors are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. The largest common factor is 72, which is our GCD. This GCD helps us reduce the fraction accurately by simplifying both numerator and denominator using the same factor.

reducing fractions

Reducing fractions is the process of simplifying a fraction to its lowest terms. Once the GCD is found, the next step is to divide both the numerator and the denominator by this number.

For the fraction \(\frac{144}{216}\), we found that their GCD is 72. Now, divide:

• 144 ÷ 72 = 2
• 216 ÷ 72 = 3

This gives us the fraction \(\frac{2}{3}\).

Reducing fractions helps in making the numbers simpler and more manageable. In this case, 144 and 216 both had a common factor (72), and dividing by this factor transformed the fraction into an easier form \(\frac{2}{3}\), representing the same ratio.

mathematical reasoning

Mathematical reasoning is crucial in verifying the steps and ensuring accuracy in reducing fractions.

After reducing the fraction to \(\frac{2}{3}\), it's important to confirm that no further simplification is possible. This is done by checking the common factors of the resultant numbers.

For \(\frac{2}{3}\), the only common divisor they share is 1, indicating that \(\frac{2}{3}\) is indeed in its simplest form.

Additionally, compare this simplified fraction to the given options to select the correct answer. In the example, \(\frac{2}{3}\) matches option B.

Using reasoning helps to confidently solve problems and verify that the solution is correct.